Power Series
Try to think
Power series = Geometric Series
.
►Refer to Math24: Power series
Power series
is actually the Geometric series
in a more general and abstract form.
![](https://solomons-mathbook.gitbook.io/~gitbook/image?url=https%3A%2F%2Fuser-images.githubusercontent.com%2F14041622%2F42147291-a00d5af4-7dff-11e8-9bbf-7ae10b740788.png&width=768&dpr=4&quality=100&sign=dc64892e&sv=1)
For easier to remember it, that could be simplified as:
![](https://solomons-mathbook.gitbook.io/~gitbook/image?url=https%3A%2F%2Fuser-images.githubusercontent.com%2F14041622%2F42147247-7517f930-7dff-11e8-8f11-e283a1ff377f.png&width=768&dpr=4&quality=100&sign=ec713418&sv=1)
In this function it's critical to know that: a_n
IS A CONSTANT NUMBER! NOT A VARIABLE !
Differentiate Power series
Differentiate Power series
►Refer to Khan academy: Differentiating power series
We have 2 ways to differentiate series, they work same way:
One way is to expand the series with real numbers (1,2,3...) and take their derivatives:
(▲Note that this is constantly true for power series.)
Another way is to calculate the term's derivative and then plug in the real numbers (1,2,3):
Either way will do, it depends on the actual equation for you to choose which way you're gonna use.
Example
Solve:
Let's organize this function to make it clear:
Example
Solve:
Notice that: If we plug in the
x=0
at beginning, everything will be0
and we don't have anything to calculate.So Let's keep the
x
in the terms until the last step.It's easier to
expand the series with real numbers
, and we're to try 3 or 4 terms in this case.Because at the end of it you'll notice, if we're doing
Third Derivative
, then more than 3 terms will just bring more0
s.First we're to organize the function in the standard power series form:
And the constant number
a_n
in the function is:Let's plug in the real number (0,1,2,3,4) for
n
to expand the series:Based on that we can take the derivatives:
Now we've got the
third derivative
, so let's plug in thex=0
and see what we get:And that's the answer.
And now you know why in this case it's a waste to calculate more terms.
Integrate Power Series
Integrate Power Series
Example
Solve:
Knowing that
Integrate a series
can be turned toa series of Integrations of terms
:Integrate the terms:
And we get a simple
Geometric series
. So that we can apply the formula of calculating geometric series:Let's apply the formula:
Integrals & derivatives of functions with known power series
► Jump over to have practice at Khan academy.
Example
Solve:
Let's assume the function of the series above is
f(x)
, and the series below isg(x)
It's clear to see that the
g(x) is the
Antiderivativeof the
f(x)`.So we just need to integrate the function of the
f(x)
:We see that the
antiderivative
can represent theg(x)
, but only with theC
in it:So we need to solve for
C
. The easiest way is to plug in0
forg(x)
:Then the answer is:
Example
Set the two series as
f(x)
andg(x)
.It's clear that
g(x) = -f'(x)
.So by differentiate
f(x)
we will get:Then
-f'(x)
would be:
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